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Computer Science > Cryptography and Security

Abstract: We prove a tight lower bound (up to constant factors) on the sample
complexity of any non-interactive local differentially private protocol for
optimizing a linear function over the simplex. This lower bound also implies a
tight lower bound (again, up to constant factors) on the sample complexity of
any non-interactive local differentially private protocol implementing the
exponential mechanism. These results reveal that any local protocol for these
problems has exponentially worse dependence on the dimension than corresponding
algorithms in the central model. Previously, Kasiviswanathan et al. (FOCS 2008)
proved an exponential separation between local and central model algorithms for
PAC learning the class of parity functions. In contrast, our lower bound are
quantitatively tight, apply to a simple and natural class of linear
optimization problems, and our techniques are arguably simpler.